If (REF ) is quantitatively well posed, then it follows from a fixed-point argument (see [1]}) that (REF ) is analytically locally well posed. In particular, there exist \(C_{0},\varepsilon _{0}>0\)  such that, for all \((f,g)\in B_{(X,Y)}(0,\varepsilon _{0})\) , there exists a unique solution \(u[f,g]\in B_{S}(0,C_{0}\varepsilon _{0})\)  to (REF ). Moreover, the map \((f,g)\mapsto u[f,g]\)  is Lipschitz continuous from \(B_{(X,Y)}(0,\varepsilon _{0})\)  to \(B_{S}(0,C_{0}\varepsilon _{0})\) , and one can expand \(u[f,g]\)  in terms of its Picard iterates. That is, define the nonlinear maps \(A_m: X \times Y \rightarrow S\)  recursively:
\(\begin{split}A_1(f,g) &:= L(f,g), \\A_m(f,g) &:= \sum _{\begin{array}{c} m_1,m_2,m_3 \ge 1: \\ m_1 + m_2 + m_3 = m \end{array}} N_3(A_{m_1}(f,g),A_{m_2}(f,g),A_{m_3}(f,g))\quad \text{for } m > 1.\end{split}\) 
